In the third part of the series presenting the Optical Gravitational Lensing Experiment (OGLE) microlensing studies of the dark matter halo compact objects (MACHOs), we describe results of the OGLE-III monitoring of the Large Magellanic Cloud (LMC). This unprecedented data set contains almost continuous photometric coverage over 8 years of about 35 million objects spread over 40 deg2. We report a detection of two candidate microlensing events found with the automated pipeline and an additional two, less probable, candidate events found manually. The optical depth derived for the two main candidates was calculated following a detailed blending examination and detection efficiency determination and was found to be τ= (0.16 ± 0.12) × 10−7. If the microlensing signal we observe originates from MACHOs, then it means their masses are around 0.2 M⊙ and they comprise only f= 3 ± 2 per cent of the mass of the Galactic halo. However, the more likely explanation of our detections does not involve dark matter compact objects at all and relies on the natural effect of self-lensing of LMC stars by LMC lenses. In such a scenario, we can almost completely rule out MACHOs in the subsolar mass range with an upper limit at f < 7 per cent reaching its minimum of f < 4 per cent at M= 0.1 M⊙. For masses around M= 10 M⊙, the constraints on the MACHOs are more lenient with f∼ 20 per cent. Owing to limitations of the survey, there is no reasonable limit found for heavier masses, leaving only a tiny window of mass spectrum still available for dark matter compact objects.